Finite Group Databasebeta

← #449458

Subgroups — #449458

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15152C211yesmaximal, characteristic
2#13911C211yesmaximal, characteristic
3#11324C212yesmaximal
4#11324C212yesmaximal
5#11257C212yesmaximal
6#11256C212yesmaximal
7#10908C212yesmaximal
8#10875C211yesmaximal, characteristic
9#10863C212yesmaximal
10C22×#107C2212yes—
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×#97C2211yescharacteristic
14C2×#400—21no—
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#382—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28#1445—21no—
29#1442—21no—
30#1442—21no—
31#1442—21no—
32#1441—21no—
33C4⋊#95—21no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×#107—24no—
56C2×#107—42no—
57C2×#107—42no—
58C2×#107—42no—
59C2×#107—42no—
60C2×#107—42no—
61C2×#106—24no—
62C2×#106—24no—
63C2×#106—42no—
64C2×#97—24no—
65C2×#97—41no—
66#400—44no—
67#384—41no—
68#384—41no—
69#384—41no—
70#384—41no—
71#382—44no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—41no—
74C4×C4⋊C4—41no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D411yescharacteristic
87C2×#95D411yescharacteristic
88#340—24no—
89#340—24no—
90#340—24no—
91#340—42no—
92#340—42no—
93#336—24no—
94#336—41no—
95#336Q818yes—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8C2×Q814yes—
118C22×C8C2×Q814yes—
119C22×C8#4114yes—
120C22×C8#4114yes—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4#4114yes—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—24no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C2×Q814yes—
176#107—48no—
177#107—48no—
178#106—48no—
179#97—44no—
180#95—24no—
181#95—41no—
182#95—41no—
183#95—44no—
184#95—42no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—42no—
189#95—82no—
190#95—82no—
191#95—82no—
192#95—82no—
193C24C22×D411yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C4⋊Q812yes—
249C22×C4C4⋊Q812yes—
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4#13712yes—
253C22×C4#13712yes—
254C22×C4C22×Q812yes—
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4H(5,𝔽2)11yescharacteristic
259C22×C4H(5,𝔽2)12yes—
260C22×C4H(5,𝔽2)11yescharacteristic
261C22×C42−1+412yes—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8#12518yes—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C42—24no—
299C42—24no—
300C42—48no—
301C42—42no—
302C42—42no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C23#52511yescharacteristic
308C23C2×C4⋊Q811yescharacteristic
309C23#53411yescharacteristic
310C23#54712yes—
311C23#55012yes—
312C23C2×AGL(1,ℤ/8)11yescharacteristic
313C23C2×AGL(1,ℤ/8)11yescharacteristic
314C23C2×#13711yescharacteristic
315C23C2×#13711yescharacteristic
316C23#57612yes—
317C23#57812yes—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#50114yes—
378C2×C4C2×#12512yes—
379C2×C4#52912yes—
380C2×C4#53812yes—
381C2×C4#54212yes—
382C2×C4#54212yes—
383C2×C4#54412yes—
384C2×C4#55111yescharacteristic
385C2×C4#55112yes—
386C2×C4#55212yes—
387C2×C4#55412yes—
388C2×C4#55412yes—
389C2×C4#55412yes—
390C2×C4#55511yescharacteristic
391C2×C4#55712yes—
392C2×C4H(3,𝔽4)11yescharacteristic
393C2×C4#56311yescharacteristic
394C8—28no—
395C8—44no—
396C22#247912yes—
397C22#255212yes—
398C22#258311yescharacteristic
399C22#271711yescharacteristic
400C22#273311yescharacteristic
401C22#311011yescharacteristic
402C22#311211yescharacteristic
403C22#312412yes—
404C22C2×#50111yescharacteristic
405C22#316412yes—
406C22#316511yescharacteristic
407C22#319211yescharacteristic
408C22#319511yescharacteristic
409C22#319911yescharacteristic
410C22#320211yescharacteristic
411C22#334312yes—
412C22#334612yes—
413C22#335312yes—
414C22#335712yes—
415C22#339812yes—
416C22#339812yes—
417C22#339912yes—
418C22#339912yes—
419C4—24no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—81no—
424C4—81no—
425C4—84no—
426C4—162no—
427C4—162no—
428C4—162no—
429C4#250512yes—
430C4#250812yes—
431C4#276312yes—
432C4#277412yes—
433C2#1594611yescharacteristic
434C2#2750312yes—
435C2#2750912yes—
436C2#2883512yes—
437C2#2883812yes—
438C2#2930211yescharacteristic
439C2#2931011yescharacteristic
440C2#3226811yescharacteristic
441C2#3227111yescharacteristic
442C2#3267911yescharacteristic
443C2#3268111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#151522
#139112
#113242
#112572
#112562
#109082
#108752
#108632